From a computational point of view, \(FBA\)  has the advantage that, if the proposed objective function is formulated as a linear function over a convex space, the optimum flux configurations can be found efficiently using Linear Programming [1]}.
Typically, \(FBA\)  formulations [2]}, [3]}, [4]}, [5]} do not make the explicit distinction between population and single-cell level metabolic spaces.
All constraints are applied over a unique space.
In the case of our two spaces formalism this is equivalent to making \(\mathbb {V}\equiv \bar{\mathbb {V}}\) , which hides an implicit culture homogeneity assumption.
This can be justified because generally, the goal is just to infer an observable flux configuration \(\bar{{v}}\)  which optimizes the objective function and a convex \(\bar{\mathbb {V}}\)  is not affected by such an assumption.
The problem to solve can then be stated as:
\(\text{optimize arg}_{\bar{{v}}} ~ f(\bar{{v}}) \\\text{subject to:~} \bar{{v}}\in \bar{\mathbb {V}}\nonumber \) 
